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A theater owner knows that to make a profit as well as to comply with fire regulations, the number of tickets that he sells can differ from 225 by no more than $75 .$ How many tickets can the theater owner sell in order to make a profit and comply with fire regulations?

Between 150 and 300 tickets inclusive

Algebra

Chapter 1

THE INTEGERS

Section 4

Solving Absolute Value Equations and Inequalities

The Integers

Equations and Inequalities

Polynomials

McMaster University

Baylor University

University of Michigan - Ann Arbor

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Okay, so we have a theater owner that needs to comply with fire regulations. And it says that that the difference between the number of tickets he sells and 2 25 can never be more than 75. And what that means is, in order to sell a prophet he need, he can't sell 75 less than 2 25 But in order to comply with fire regulations, he can only sell up to 75 more than 2 25 because otherwise the theater would get too crowded. So what this is saying is that the distance from 75 from 2 25 is it is. It's an absolute value. So the distance between 2 25 and the amount of tickets I sell has to be less than or equal to 75 and I want to make sure it's less than or equal to 75. It is, and so that's the absolute value when you say said, and now it wants to know how many tickets does the theater owner have to sell in order to make a profit and at the same time comply with fire regulations? And so now we just need to solve for the number of tickets here, which is X in this case. And so we split our equation, since the absolute values already isolated to 25 minus X is less than or equal to 75 2 25 minus X is greater than or equal to minus 75. And so to solve this one, we just subtract 2 25 75 minus 2 25 is minus 1 50 So x is less than or equal to minus 1 50 Here we divide by minus on. And so we have to flip our sign. So we have X is bigger than 1 50 and we did the same thing here. Subtract 2 25 over. And so we're gonna have minus X is greater than our equal to minus 300. And then we divide by negative one, which means we have to flip our sign. So we have X is less than or equal to 300. And so what he needs to do is he needs to have all of this ticket sales be in the range of 1 50 2 300 with 2 25 being in the middle. And that should make sense because what's 75 away from 2 25? 300? So he can go that far away. Up 300. He's just barely meeting his fire regulations, his theaters just about to be overcrowded. But if he only sells 1 50 then he's probably breaking even, or just barely making a profit. And so there's this range of 150 numbers of tickets take itself, and this is the range where the theater is making a profit.

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